Bulletin of the Seismological Society of America, Vol. 98, No. 4, pp. 1633–1641, August 2008, doi: 10.1785/0120070191
Slip-Length Scaling in Large Earthquakes: The Role of Deep-Penetrating
Slip below the Seismogenic Layer
by Bruce E. Shaw and Steven G. Wesnousky
Abstract
Coseismic slip is observed to increase with earthquake rupture length for
lengths far beyond the length scale set by the seismogenic layer. The observation,
when interpreted within the realm of static dislocation theory and the imposed limit
that slip be confined to the seismogenic layer, implies that earthquake stress drop
increases as a function of rupture length for large earthquakes and, hence, that large
earthquakes differ from small earthquakes. Here, a three-dimensional elastodynamic
model is applied to show that the observed increase in coseismic slip with rupture
length may be satisfied while maintaining a constant stress drop across the entire spectrum of earthquake sizes when slip is allowed to penetrate below the seismogenic layer
into an underlying zone characterized by velocity-strengthening behavior. Is this deep
coseismic slip happening during large earthquakes? We point to a number of additional associated features of the model behavior that are potentially observable in the
Earth. These include the predictions that a substantial fraction, on the order of onethird of the total coseismic moment, is due to slip below the seismogenic layer and that
slip below the seismogenic layer should be characterized by long rise times and a
dearth of high-frequency motion.
Introduction
Scholz (1982) pointed out many years ago the observation that average slip in large earthquakes continued to increase with rupture length well beyond the seismogenic
crust depth. He noted this observation was puzzling because
it was expected that ruptures broke dynamically only down
to the bottom of the seismogenic depth, and this finite depth
would be expected to saturate the slip for very large events
if the constant stress-drop scaling for small earthquakes
(Hanks, 1977) continued to hold for the large earthquakes.
A debate ensued about whether the largest earthquakes continued to have even larger slip (Scholz, 1982; Pegler and Das,
1996; Henry and Das, 2001) or the slip eventually saturated
(Romanowicz, 1992, 1994; Scholz, 1994; Bodin and Brune,
1996) and about whether longer length scales (Romanowicz,
1992; Scholz, 1994) or multiple length scales (Manighetti
et al., 2007) might be entering into the problem. This debate
has important implications for earthquake hazard analysis
because it is the largest events that dominate the net slip
along faults, and this plays a role in determining how often
the large events will occur as well as how strong the shaking
will be in these events. The empirical scaling of slip versus
area of rupture (a related but slightly different question from
slip versus length scaling) plays a central role in current
earthquake hazard estimates, and differences in assumed
scaling have significant impacts on the estimates (Wells
and Coppersmith, 1994; Hanks and Bakun, 2002; Working
Group on California Earthquake Probabilities (WGCEP),
2003, 2007). Understanding the physical basis of the sliplength scaling would help to better understand which
moment-area scalings to use and how best to extrapolate
them to the largest events.
A central question arising from the observations was
whether differing physical processes were needed to account
for the differences between the large and small earthquakes in the empirical scaling of slip and inferred stress
drop. Shaw and Scholz (2001) found that numerical simulations of scale invariant physics were able to match the observations of slip-length scaling both in terms of the mean
and the scatter, suggesting that new unaccounted for physical
processes were not needed. Their results reaffirmed the
surprising feature of the data that the slip of the largest
events continued to increase at length scales much longer
than the seismogenic depth. This puzzle has led to recent
attempts at an explanation, including one involving multiple
length scales based on fault segmentation (Manighetti et al.,
2007). Building on an idea originally proposed by Das
(1982), King and Wesnousky (2007) showed that the apparent conflict with static elastic dislocation theory (increase in
stress drop) implied by the increase of slip with length for
large earthquakes is resolved if some significant fraction
of coseismic slip is allowed to extend below the base of
the seismogenic layer into a medium of stable sliding that
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B. E. Shaw and S. G. Wesnousky
remains stressed at or close to failure. Here, we look in more
detail at the three-dimensional elastodynamic model studied
by Shaw and Scholz (2001) and find that slip in the model in
the largest events is indeed penetrating deep into the stably
sliding lower fault. The penetration of rapid coseismic slip
into the underlying zone of stable slip decreases the effective
vertical stiffness and thereby allows additional slip in the
seismogenic layer. Thus, we find a physical basis for the puzzling scaling seen in the numerical simulations that match
well the observations, and we find that the deep slip hypothesis in the Das (1982) and King and Wesnousky (2007)
explanations is indeed being chosen dynamically by a physical model. This poses a key question for earthquakes: is
slip penetrating coseismically much deeper than is typically
assumed in seismological, geodetic, and seismic hazard
models, and how might we determine that?
The Model
The model bulk consists of a three-dimensional scalar
wave equation (mode III elastic). The rectangular bulk is
loaded at constant velocity on a far boundary parallel to a
vertical fault. At the top and bottom of the bulk we have free
boundary conditions, and in the direction along the length of
the fault we use periodic boundary conditions. Close to the
far loading boundary, we use a viscous layer to damp out
waves and minimize reflections. On the fault we use a frictional boundary condition with the strain being equal to the
frictional stress on it. The friction on the fault is the source of
all the nonlinearity in the problem and is the key to all the
interesting complex behavior.
Two kinds of friction are used on the fault. At depth it is
stably sliding velocity-strengthening friction. Above that, in
a seismogenic layer extending from the top down to the seismogenic depth, it is unstably sliding with a mixture of slip
and velocity-weakening friction (Shaw and Rice, 2000;
Shaw and Scholz, 2001; see the Appendix for more model
details). This seismogenic depth H sets the fundamental
length scale in the problem, which we scale all lengths in the
problem to and thus define as unity, a length scale in the real
earthquake problem corresponding to around 15 km for vertical strike-slip faults (for dipping faults, the relevant length
is the down-dip fault width, so H is much wider for shallow
dipping subduction zone events). When we run the model for
a long time, it settles down (after a few large event cycles)
onto an attracting statistically steady state. For a range of frictional weakening parameters in the problem (see the Appendix for details), we get a wide range of sizes of the large
events that do not break the whole length of the fault and
on which we can thus examine slip-length scaling issues.
The scaling results we present in the following discussion
appear to be quite insensitive to essentially all of the parameters in the problem, when we operate with large domains
and are in the regime where we get a range of sizes of large
events.
Results
Figure 1 shows the scaling of mean surface slip D with
surface rupture length L in the model. Only events that break
the surface are shown. The individual circles are for individual model events. Shaw and Scholz (2001) demonstrated
the remarkable similarity of this measurement in the model
as compared with earthquake data for large strike-slip events,
not only in terms of the average behavior but also the scatter
around the average. The solid line on this plot shows the scaling proposed by Shaw and Scholz (2001) for a constant
stress-drop model that takes into account the finite width
H of the seismogenic layer and the free surface at the top:
D∼
8
< 111
L ≤ 2H;
: 1 1 1
L 2H
L > 2H:
LL
(1)
The dashed line shows this same scaling law where we have
now added a scaling factor λ to increase the seismogenic
width H to have an effective mechanical width W, so
now W λH, and H in equation (1) is replaced by λH.
In the figure, we have also rescaled the amplitude, so the
asymptotic slip for very large events is the same for the
two curves. The solid line scaling in equation (1) with H assumes the slip extends only to the bottom of the seismogenic
zone. Manighetti et al. (2007) allowed for a λ scaling related
to rupture length to better empirically fit the observed data.
Here, we consider the generalized λ scaling based on the increased penetration depth of slip, leading to a larger effective
width. In Figure 1 we show a fit for a mechanical width W
that is an effective tripling of H, using λ 3. As can be seen
Figure 1.
Average surface slip D versus rupture length L for
large surface rupturing events in a three-dimensional scalar elastodynamic model. Length L is scaled by the seismogenic width H
showing the aspect ratio of the ruptures. Individual circles are
for individual rupture events. The solid curve is with H as in equation (1). The dashed line shows fit for larger W 3H.
Slip-Length Scaling in Large Earthquakes
by eye, this rescaled W provides a much better fit to the
model data. Manighetti et al. (2007) find the best-fitting
values of λ ≈ 3 for their earthquake data set; they argue,
however, for multiple integer values of λ based on fault segmentation. Instead, in what follows, we show the rescaled W
has a basis, at least in our numerical models, in the way the
slip penetrates deeply coseismically into the stably sliding
lower fault below the seismogenic zone.
Comparing Figure 1 with observations, the longest
model events are much longer than the longest observed
events; the largest observed aspect ratios for earthquakes,
which occur in very long strike-slip events, have
L ∼ 400 km, which for H 15 km gives an L=H ∼ 25 or
30. Our largest aspect ratios of 80 here in the model are
thus much longer than the longest observed events. This
helps, however, to elucidate the asymptotic behavior for very
long events, and it also helps to explain why saturation of slip
in the observations is barely, if at all, seen: at aspect ratio 30
the model is also just barely beginning to saturate in slip.
Figure 2 illustrates the key behavior we are observing
from a seismic point of view of slip penetrating coseismically
quite deep. We plot a vertical array of velocity seismometers
placed on the model fault, going from the surface to well
below the seismogenic layer and plotting the velocities as
a function of time, for an example large model event. To
aid in visualizing the curves, a vertical offset proportional
to the depth of the velocity meter is added. The array is located in a narrow vertical strip surrounding the epicenter of
this example large event. Only nonzero velocities are shown.
We see a key difference for behavior between the seismogenic upper fault, which slips with both long-period motion
and lots of high frequencies, which shows in the very jagged
behavior of the velocity curves as a function of time, and the
stably sliding lower fault, which slips much more smoothly
and with mainly long-period motion. Note also the time delay in rapid slip as the slip pushes deeper and deeper into the
stably sliding layer, with the leading rupture front propagating through the seismogenic layer. Note as well, however,
that this time delay is not large compared to the rupture timescale: these significant motions occurring on the stably sliding deep fault are happening coseismically. Indeed, while
there is some afterslip, we are getting substantial deep coseismic slip, and it is this deep coseismic slip that is feeding back
mechanically into the slip in the seismogenic layer. One other
important point can be made from this plot: the deeper coseismic slip may not be easy to see in seismological inversions of fault slip, which are much more sensitive to the
broad spectrum radiating seismogenic layer than the lowfrequency deeper layers, but they would be a key signal
to look for to image this behavior.
How significant is this lower frequency slip in the deeper
layers? Figure 3 shows a plot of the ratio of the coseismic
moment below the seismogenic layer to the moment in
the seismogenic layer. We see that something on the order
of one-half of the seismogenic moment, or one-third of
the total moment, is occurring in this deeper stably sliding
1635
Figure 2. Velocity as a function of time for an example large
event on an array of velocity meters. The array is placed along strike
around the epicenter, down depth of the fault. For ease in visualization, velocity is offset proportional to depth z, with the offset equal
to 0:1z=H. At each depth five velocity meters laterally are displaced
at spacing H=6 around the hypocenter in the along-strike direction;
the width of the line gives a sense of spread in velocities along strike
near the epicenter. In the depth direction, 14 of these lateral velocity
meter lines are shown. Note high-frequency motions at shallow
depths, in addition to the low-frequency component, while motion
at depth is mainly low frequency. Only nonzero velocities are
plotted. Note the delay of the onset of rapid motion at increased
depths, as slip is driven deeper, but also note substantial coseismic
slip on stably sliding lower fault. Units are dimensionless; for time,
it is the time is takes a wave propagating at the shear-wave velocity
to propagate the seismogenic crust depth H; thus, a time of unity
corresponds to around 5 sec for a 3 km=sec shear wave propagating
across 15 km. For velocity, the unit corresponds to the stress drop
divided by the shear modulus times the shear velocity, a number
around 0:3 m=sec for typical Earth values (see the appendix in
Shaw [2006]) for full dimensionfull conversions).
Figure 3.
Moment in the stable sliding deeper layer divided by
moment in the unstably sliding seismogenic layer. Values on the
order of one-half of the seismogenic moment, or on the order of
one-third of the total moment, are found for the very large events.
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B. E. Shaw and S. G. Wesnousky
Figure 4. Slip averaged along strike as a function of depth for individual events, shown with thick solid lines. (a) Vertical axis is logarithmic scale; horizontal axis is linear scale. Thin dashed lines show exponential fit to fall-off of slip with depth below the seismogenic layer,
with fit between depths H and 4H. The thin lines show an exponential decay scale H for comparison. (b) Linear scales on vertical and
horizontal axes. Thin dashed lines show linear fit to fall-off of slip with depth around the bottom of the seismogenic layer. The thin line
shows a linear fall-off that has a slip that extrapolates to zero at the depth W 2H.
layer (comparable fractions of the moment occur for other
friction parameter values, though the exact numbers do depend somewhat on the friction parameters). This suggests
one potential way of detecting this deeper slip: inversions
sensitive to different frequency bands would place the
low-frequency moment release at locations that differ from
higher frequency motion. This approach is not without
complications, however, as directivity effects can also shift
centroids of different frequency bands.
A plot of the slip as a function of depth, Figure 4, gives
an indication of what the deep slip looks like. Figure 4 shows
for each large model event the slip averaged along strike
plotted as a function of depth. The depth unity marks the
Figure 5.
Depth W where slip extrapolates to zero slip from the
slip-depth profile, as fit in Figure 4b, as a function of rupture length
L. Note the roughly doubling of the effective width W relative to
seismogenic width H for large aspect ratio L=H.
transition from the velocity-weakening seismogenic to the
velocity-strengthening stably sliding lower fault. We plot
the log of the slip on the vertical axis against the linear depth
on the horizontal axis in Figure 4a. We see that the slip is
well fit by an exponential decay into the stably sliding layer.
To best describe the mechanical impact of the deep slip, we
fit a linear function to the slip around the seismogenic depth,
shown in Figure 4b. Extrapolating this linear fit down to the
depth where the slip would be zero gives an effective width
W. While these slip profiles are not simple self-similar profiles as postulated in King and Wesnousky (2007), the deep
extended slip does play an analogous important mechanical
role, as we will see.
Figure 5 shows these effective W length scales plotted as
a function of the length of the rupture. We see a significant
increase of W above H for ruptures with L longer than a few
H, and some continued increase for larger L. The small continued increase of W with L plays out into a somewhat larger
best-fitting single value of λ than the individual values of W
might suggest: if we do a least-squares fit for the data for a
best-fitting λ in Figure 1, we get λ 4:1, which is larger
than the values of W found in Figure 5. This is an effect
caused by the increasing W dependence of L. This has important implications for inferring W from a best-fitting value
from observational data of λ ≈ 3: it does not mean that
W=H ≈ 3 necessarily in the Earth, but rather it could mean
W=H < 3 with W increasing somewhat with L.
That these W are indeed impacting the seismogenic slip
mechanically can be seen when we correct individual events
based on their individual W. Figure 6 shows the estimated
mean stress drops for the same model events as shown in
Figure 1, where we take into account the length scales of
the ruptures and compare the estimated stress drops with
the average of the stress drops measured directly on the
Slip-Length Scaling in Large Earthquakes
1637
values from slip and rupture lengths are completely independent, and thus, the tracking of the measurement fluctuations
up and down by the estimates indicates we are capturing a
key part of the process and provides additional support for
the methodology.
In Figure 6 the stress drops appear nearly constant
across all length scales. This is not always the case for all
friction parameter values—there can, for example, be some
slight increase in stress drop with increased rupture length.
What is most robust, however, is that the stress drops that do
occur are best tracked by W rather than H, and it is the W
dependence on L along with any potential stress-drop
changes that is needed to explain the slip-length scaling.
Comparison with Observations
Figure 6.
Average stress drop Δσ versus rupture length L for
events in Figure 1. Measured values of mean stress drop in events
are shown with crosses (×). Estimated values when penetration
depth W is taken into account are shown with pluses (+). Circles
show estimates of stress drop for the same events but assuming that
the depth of rupture is fixed to the depth of the seismogenic layer H.
Note effective W is much better fit to measured values than H is.
seismogenic rupture areas (that is, areas within the seismogenic zone with depth z < H that ruptures); these measured
average stresses are shown with the cross symbol (×).
To estimate the stress drops of the large events (large events
are defined here as those with seismogenic rupture area
A > 2H2 ), we estimate the along-strike lengths using A=H,
where A for real earthquakes would be determined from
aftershock distributions. (Using A=H as an estimate for
the along-strike length instead of L makes little difference
for the very large events, but it is a slightly better estimate
for events that barely rupture the surface.) For the width
of the ruptures we examine estimates of stress drop Δσ using
W and H. Based on the Knopoff (1958) solution but modifying for finite A, we obtain
1
1
:
Δσ D
A=H 2W
(2)
(The modulus here is dimensionless, scaled to unity.) As can
be seen in Figure 6, the estimates using W, shown with the
plus symbols (+), are quite good—generally within 10 or a
few tens of percent, with, importantly, no bias with L. They
are slightly systematically low, but this is not very important
in that the prefactor depends on a very simple assumption
about the slip geometry, equation (2). What is most important
is that there is no systematic dependence on L. Note as well
the estimates are much superior to those using H, shown with
the circle symbol, with the pluses from W falling much closer
to the directly measured cross values than the circle estimates
using H; moreover, the circle H estimates also have an undesirable systematic error dependence on L. Note that the
direct measurements of stress drop and the estimated average
While the prevailing view is that significant coseismic
slip is mainly restricted to the seismogenic layer, with perhaps some slight penetration below that over small distances,
we can ask whether the observations require this. Global
Positioning System (GPS) observations generally have little
depth resolution (Thatcher et al., 1997), especially below the
upper 5 km or so on vertical strike-slip faults even under
the best circumstances (M. Page et al., unpublished manuscript, 2007), and need to be in place beforehand to distinguish coseismic slip from afterslip. King and Wesnousky
(2007) discuss GPS constraints in more detail and infer that
deep coseismic slip is not ruled out by current observations.
Seismic inversions for slip also suffer greatly from spatial resolution problems (Beresnev, 2003). While they are relatively robust at reconstructing moment rate of the rupture,
mapping this spatially onto the fault suffers from a tremendous nonuniqueness problem and is reflected in the huge
variability in mapped slip of the same large ruptures from
different groups (Mai, 2007), as, for example, the different
inversions for the 1992 Mw 7.3 Landers earthquake (Cohee
and Beroza, 1994; Wald and Heaton, 1994; Cotton and
Campillo, 1995; Zeng and Anderson, 1996; Hernandez et al.,
1999). Many inversions do seem to prefer to add slip deeper
than is often considered, evidenced by slip piling up at the
bottom edges of the inversions (for example, for Landers
[Wald and Heaton, 1994; Cotton and Campillo, 1995]).
But this also reflects a lack of sensitivity at these depths,
so one should be cautious to not overinterpret this.
When we have both seismic and GPS measurements
available, while they often agree well on total moment, they
also typically have substantial differences in where they ascribe slip (for example, Konca et al. [2007]). This illustrates
well our main point about the observational constraints on
potentially deep coseismic slip: we do not expect that the slip
is evading detection, rather that there are insufficient constraints on where it is happening. It is not missing slip,
but misplaced slip we are looking for.
Aftershocks do a remarkably good job of illuminating
the rupture area in the seismogenic region. But at depths below the seismogenic zone, the stably sliding lower fault is
1638
B. E. Shaw and S. G. Wesnousky
conditionally stable, potentially supporting ruptures into it
but not initiating ruptures, and thus aftershock depths do
not constrain the depth extent of slip for large events.
Figure 2 motivates a look at depth dependent differences
between high-frequency mapping of fault slip and lowfrequency inversions. In the case of the great M 9.1 2004
Sumatra tsunami earthquake, the high-frequency centroid
(Ishii et al., 2005) tends to be deeper than the moment centroid (Tsai et al., 2005), likely due to extensive slip in stable
sliding nonseismogenic shallower layers. Thus, while the
effect we describe of differences in location between broadband and low-frequency slip appears to be occurring and
measurable, the shallow stably sliding parts seem to be dominating, in terms of moment centroid, a potential contribution
due to the deeper stably sliding parts.
Conclusion
We have shown in a three-dimensional elastodynamic
model that significant coseismic slip can penetrate deep into
the stably sliding fault below the seismogenic layer during
very large events. This deeper slip occurs with little highfrequency radiation, and thus it would be easy to mislocate
seismologically. It occurs despite the fact that slip in the
stably sliding layer is an energy sink but as a consequence
accommodates additional slip through a reduced effective vertical stiffness in the upper seismogenic layer. In
the model, on the order of one-half the seismogenic moment,
or one-third the total moment, occurs in the deeper stably
sliding lower layer. The observed exponential fall-off of slip
with depth in these model events plays a mechanical role
of increasing the effective coseismic depth. This increased
effective depth matches well our observed slip versus length
scaling data and matches the measured stress drops in the
model. The key questions now are whether this same
mechanism is operating in the Earth and whether significant
coseismic slip is occurring below the seismogenic layer.
Acknowledgments
One of us (B.E.S.) thanks Barbara Romanowicz for discussions that
stimulated this work. One of us (S.G.W.) benefited greatly from the insight
of G. C. P. King in the development of ideas bearing on deep coseismic slip.
This work is partially supported by National Science Foundation Grant
Number EAR03-37226 and by the Southern California Earthquake Center
(SCEC).
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on the fault. At a far boundary opposite and parallel to the
fault, we drive the boundary at a constant slow plate velocity
ν, with the boundary condition
∂U ν:
∂t yLy
At the top and bottom of the bulk we have free boundary
conditions:
∂U ∂U 0:
(A3)
∂z z0 ∂z zLz
Along the fault direction, we use periodic boundary conditions:
Ux Lx Ux:
(A4)
A viscous boundary layer near the driving boundary damps
waves and minimizes reflections. There, the bulk generalizes
to
∂2U
∂U
∇2 U ∇ · ξy∇
∂t
∂t2
(A5)
with ξy y y0 ξ 0 Hy y0 linearly increasing from
zero (H is the Heaviside step function).
All of the interesting behavior comes from the frictional
boundary condition on the fault, which contains all of the
nonlinearity in the problem. On the fault, the strain equals
the frictional traction stress Φ:
∂U Φ:
∂y y0
Appendix
(A6)
Friction
We use a stick-slip friction, with a stabilizing viscous
term along strike:
Model
Φϕ
Model Equations
The bulk equations are a three-dimensional scalar wave
equation (mode III elasticity) for the displacement field U:
∂2U
∇2 U;
∂t2
(A2)
(A1)
where t is time and ∇2 is the three-dimensional Laplacian
operator ∇2 ∂ 2 =∂x2 ∂ 2 =∂y2 ∂ 2 =∂z2. We use a rectangular geometry, with the fault on a vertical plane, with
x the direction along strike of the fault, y the direction perpendicular to the fault, and z the depth direction, down-dip
∂S 0
∂S
∂S
η∇2jj
:
;
t
≤
t
H
∂t0
∂t
∂t
(A7)
∂U
Here, ∂S
∂t ∂t jy0 is the slip rate on the fault, with ϕ depending on the past history of slip. The function H is the antisymmetric step function, with
8
< ∂bS
∂S
∂t ≠ 0;
(A8)
H ∂t
: jHj < 1 ∂S 0;
∂t
b
where ∂∂tS is the unit vector in the sliding direction. Thus, H
represents the stick-slip nature of the friction, being multivalued at zero slip rate.
1640
B. E. Shaw and S. G. Wesnousky
The parameter η is a constant that sets the amount of
viscosity and sets the scale at which the small wavelengths
are stabilized. The subscript on the viscous term Laplacian
denotes that it is the derivative parallel to the fault, which
gives ∇2jj ∂ 2 =∂x2 ∂ 2 =∂z2 for the geometry we consider
here (Langer and Nakanishi, 1993).
Depth Dependence of Friction
We consider a fault that has two different behaviors: an
unstable sliding frictional weakening seismogenic layer from
the surface down to a depth H and a stable sliding frictional
strengthening layer below that. In the strengthening layer,
ϕjz>H Φ0 a
∂S
;
∂t
(A9)
where a is the slope of the velocity-strengthening term. Φ0 is
a constant that sets the sticking threshold. It turns out to be an
irrelevant parameter in the problem, as long as it is large
compared to the maximum friction drop, so as to prevent
backslipping.
In the weakening layer, we use a mixture of slip and
velocity weakening, based on a frictional heating mechanism
(Shaw, 1995, 1998; Shaw and Rice, 2000):
ϕj0≤z≤H Φ0 αQ
:
1 αQ
(A10)
The second term in equation (A10) contains the key dependence on slip and slip rate in the friction, through the variable
Q. The variable Q is something like heat, which accumulates
with increasing slip rate and dissipates on a timescale 1=γ:
∂S ∂Q
γQ :
(A11)
∂t
∂t
The dissipation with γ gives a simple physically motivated
healing mechanism, which also turns out to give a nice range
of properties. An equivalent integral solution of Q,
Z
Qt t
∞
0 ∂S
eγtt 0 dt0 ;
∂t
(A12)
shows that, when γ is small compared to unity, the inverse
rupture timescale, Q is just the slip in an event, while when
γ ≫ 1, Q is 1=γ times the slip rate. When γ ∼ 1, as we will
often use in our three-dimensional simulations, we get a
mixture of slip and velocity weakening.
The constant α sets the slope of the stress drop with heat
Q. This parameter plays a crucial role in the problem and is a
key control parameter.
This heat-weakening friction is a simple mathematical
representation (Shaw, 1998) of a physical idea that goes back
to Sibson (1973), who considered how frictional heating
raised the temperature and pressure of pore fluids, thereby
decreasing the effective normal stress and thus inducing frictional weakening from frictional heating. Shaw (1995) presented simplified self-consistent dynamics of this effect,
showing that one got slip weakening and velocity weakening
as end-member limits, depending on whether the dissipation
of heat was slow or fast, respectively. Earthquakes would be
able to dissipate excess pressure with an elastic expansion
mode (Mase and Smith, 1987), a mode that can happen
on the fast rupture timescale. This fast relaxation mechanism
suggests values of γ on the order of unity or larger as
the most appropriate values. Thus, a mixture of slip- and
velocity-weakening effects are likely occurring. We typically
operate in this range because of the physical motivation, but
importantly, we also find that, unlike in lower dimensional
models where slip weakening alone gives complexity (Shaw,
1995, 1998), in three dimensions we need some velocity
weakening as well to get complex sequences of events. If
we use slip weakening alone in three dimensions we find,
for the parameters we have been able to study, the attracting
statistically steady state to be generally system spanning
events.
Parameters
A finite difference second-order approximation is made
of the bulk equations, which are solved explicitly in time.
The default parameter values in the simulations shown are
as follows. Geometry parameters δ x 1=6, δ z 1=6,
δy 1=12, Lx 80, and Ly 4; viscous boundary layer
parameters y0 Ly 1 and ξ 0 0:1; loading rate
ν 0:0003; and friction parameters α 20, γ 1, and
η 0:005 in the weakening layer and a 100 in the
strengthening layer. These parameters are chosen for the following reasons. For the grid resolution parameters δx , δz , and
δy , we would like to make these as small as possible, but
numerical costs scale as the fourth power of the grid resolution, so we are limited in how small we can make them. We
take the grid resolution along the two directions of the fault
to be equal. With δ x δ z 1=6 we are able to get a range of
sizes of small events down to a few kilometers on a side,
corresponding to roughly M 5 for real earthquakes. Because
we are most interested in very large events, this is sufficient.
The grid resolution perpendicular to the fault needs to be
even more resolved than the fault parallel direction, and a
factor of 2 has been seen to be sufficient for this additional
resolution; hence, δy 1=12. Fault parallel domain size Lx
needs to be large enough so the longest events generally do
not break the whole fault. Periodic boundary conditions are
used along the fault. Fault perpendicular domain size Ly
should be as large as possible, but numerical costs make this
choice, which is large compared to unity, large enough so
any imperfectly absorbed waves will not interfere with the
dynamics on the fault. A viscous boundary layer away from
Slip-Length Scaling in Large Earthquakes
the fault is used to damp out waves over a length scale of
unity; hence, y0 Ly 1, with a linearly increasing viscous
term up to a value of ξ0 0:1. The friction parameter α 20 is chosen in the range of values where the scaling of small
events is consistent with the observed scaling of moment
versus area for small events (Hanks, 1977); this occurs over
some finite range of α values, a range that appears to increase
as grid resolution increases (though we are quite limited in
the range of grid resolutions we can explore due to the fourth
power dependence of computational time on grid resolution).
The parameter γ 1 is chosen to get a mixture of slip weakening and velocity weakening. We find we need some velocity weakening to get complex event sequences in three
dimensions. The parameter ϵ 0:005 is chosen to stabilize
the small length scales, but not too large that it affects larger
scale events. Typical catalog lengths are νt ∼ 10 so that many
repeat times of large events are simulated, corresponding to a
timescale on the order of thousands of years. The loading rate
parameter ν 0:0003 is chosen as a compromise between
wanting small values to have a clear separation of timescales
between events but wanting a large value to have a long catalog of events.
1641
we believe this insensitivity to the velocity slope parameter
suggests a robustness of our results with respect to treatments
of this layer, comparison of our findings with models using
more sophisticated deeper rheology would be useful. Tse and
Rice (1986) studied two-dimensional quasistatic models with
logarithmic velocity strengthening at depth and found only
minimal penetration below the seismogenic layer. Lapusta
and Rice (2003) studied two-dimensional dynamic models
and again found minimal penetration depth. Full rate-andstate equations have additional potential complexities in
behavior when the full equations, rather than steady-state
approximations, are used (Perfettini and Ampuero, 2006;
A. Helmstetter and B. E. Shaw, unpublished manuscript,
2007); thus, the ability of slip to penetrate deeply below
the seismogenic layer in a rate-and-state friction framework
is an unsettled matter. Indeed, recent work subsequent to
this work (Hillers and Wesnousky, 2008) shows quasistatic
models with more gradual variations in rate-and-state parameters than used by previous models (Tse and Rice, 1986;
Lapusta and Rice, 2003) allow for much deeper penetrating
slip. Because some theoretical models clearly allow for it,
and the slip-length observations are nicely explained if this
is happening, the key question again becomes an observational one: is slip penetrating deep below the seismogenic
layer in the Earth?
Discussion
There is a remarkable insensitivity of the penetration
length scale λ to the velocity-strengthening values a: values
of a 10, 100, and 1000 lead to the same effective W. This
is additionally significant because we do not use the functional form of velocity strengthening generally expected
to hold in the deeper fault, the direct effect of logarithmics
velocity-strengthening dependence seen in laboratory measurements (Dieterich, 1979) and inferred from observations
(Marone et al., 1991; Schaff et al., 1998). Here, we argue that
the insensitivity to this velocity-strengthening parameter supports our use of the numerically simpler linear form. While
Lamont-Doherty Earth Observatory
61 Route 9W
Palisades, New York 10964
[email protected]
(B.E.S.)
Center for Neotectonic Studies
University of Nevada Reno
Reno, Nevada 89557
[email protected]
(S.G.W.)
Manuscript received 27 July 2007